Curve

CURVE (Lat. curvus, bent), a word commonly meaning a shape represented by a line bending continuously out of the straight without making
an angle, but only properly to be defined in its geometrical sense in the terms set out below. This subject is treated here from an historical point of view,
for the purpose of showing how the different leading ideas were successively arrived at and developed.

i. A curve is a line, or continuous singly infinite system of points. We consider in the first instance, and chiefly, a plane curve described according to a
law. Such a curve may be regarded geometrically as actually described, or kinematically as in the course of description by the motion of a point; in the former
point of view, it is the locus of all the points which satisfy a given condition; in the latter, it is the locus of a point moving subject to a given condition.
Thus the most simple and earliest known curve, the circle, is the locus of all the points at a given distance from a fixed centre, or else the locus of a point
moving so as to be always at a given distance from a fixed centre. (The straight line and the point are not for the moment regarded as curves.)

Next to the circle we have the conic sections, the invention of them attributed to Plato (who lived 430-347 B.C.); the original definition of them as the
sections of a cone was by the Greek geometers who studied them soon replaced by a proper definition in piano like that for the circle, viz. a conic section (or
as we now say a " conic ") is the locus of a point such that its distance from a given point, the focus, is in a given ratio to its (perpendicular) distance
from a given line, the directrix; or it is the locus of a point which moves so as always to satisfy the foregoing condition. Similarly any other property might
be used as a definition; an ellipse is the locus of a point such that the sum of its distances from two fixed points (the foci) is constant, etc., etc.

The Greek geometers invented other curves; in particular, the conchoid (q.v.), which is the locus of a point such that its distance from a given line,
measured along the line drawn through it to a fixed point, is constant; and the cissoid (q.v.), which is the locus of a point such that its distance from a
fixed point is always equal to the intercept (on the line through the fixed point) between a circle passing through the fixed point and the tangent to the
circle at the point opposite to the fixed point. Obviously the number of such geometrical or kinematical definitions is infinite. In a machine of any kind, each
point describes a curve; a simple but important instance is the " three-bar curve," or locus of a point in or rigidly connected with a bar pivoted on to two
other bars which rotate about fixed centres respectively. Every curve thus arbitrarily defined has its own properties; and there was not any principle of
classification.

2. Cartesian Co-ordinates. The principle of classification first presented itself in the Geometrie of Descartes (1637). The idea was to represent any curve
whatever by means of a relation between the co-ordinates (*, y) of a point of the curve, or say to represent the curve by means of its equation. (See GEOMETRY:
Analytical.)

Any relation whatever between (x, y) determines a curve, and conversely every curve whatever is determined by a relation between (x,y).

Observe that the distinctive feature is in the exclusive use of such determination of a curve by means of its equation. The Greek geometers were perfectly
familiar with the property of an ellipse which in the Cartesian notation is X 1 /a?+y 2 /b 2 =i, the equation of the curve; but it was as one of a number of
properties, and in no wise selected out of the others for the characteristic property of the curve.

3. Order of a Curve. We obtain from the equation the notion of an algebraical as opposed to a transcendental curve, viz. an algebraical curve is a curve
having an equation F(x, y)=o where F(x, y) is a rational and integral function of the coordinates (x, y); and in what follows we attend throughout (unless the
contrary is stated) only to such curves. The equation is sometimes given, and may conveniently be used, in an irrational form, but we always imagine it reduced
to the foregoing rational and integral form, and regard this as the equation of the curve. And we have hence the notion of a curve of a given order, viz. the
order of the curve is equal to that of the term or terms of highest order in the co-ordinates (x, y) conjointly in the equation of the curve; for instance, xy
1=0 is a curve of the second order.

It is to be noticed here that the axes of co-ordinates may be any two lines at right angles to each other whatever; and that the equation of a curve will be
different according to the selection of the axes of co-ordinates; but the order is independent of the axes, and has a determinate value for any given curve.

We hence divide curves according to their order, viz. a curve is of the first order, second order, third order, etc., according as it is represented by an
equation of the first order, ax-\-by-\-c = o, or say (*jz, y, i)=o; or by an equation of the second order, ax*+2hxy+by*+2fy-{-2gx+c=o, say (*5*, y, i) 2 =o; or
by an equation of the third order, etc.; or what is the same thing, according as the equation is linear, quadric, cubic, etc.

A curve of the first order is a right line; and conversely every right line is a curve of the first order. A curve of the second order is a conic, and is
also called a quadric curve; and conversely every conic is a curve of the second order or quadric curve. A curve of the third order is called a cubic; one of
the fourth order a quartic; and so on.

A curve of the order m has for its equation (*J#, V, i) m = o; and when the coefficients of the function are arbitrary, the curve is said to be the general
curve of the order m. The number of coefficients is %(m+i) (m+a); but there is no loss of generality if the equation be divided by one coefficient so as to
reduce the coefficient of the corresponding term to unity, hence the number of coefficients may be reckoned as %(m+i) (m+2) i, that is, $m(m+3) ; and a curve of
the order m may be made to satisfy this number of conditions; for example, to pass through \m(m+3) points.

It is to be remarked that an equation may break up; thus a quadric equation may be (ax+by+c) (a'x-\-b'y+c')=o, breaking up into the two equations ax+by+c =
o, a'x+b'y+c' = o, viz. the original equation is satisfied if either of these is satisfied. Each of these last equations represents a curve of the first order,
or right line; and the original equation represents this pair of lines, viz. the pair of lines is considered as a quadric curve. But it is an improper quadric
curve; and in speaking of curves of the second or any other given order, we frequently imply that the curve is a proper curve represented by an equation which
does not break up.

4. Intersections of Curves. The intersections of two curves are obtained by combining their equations; viz. the elimination from the two equations of y (or
x) gives for x (or y) an equation of a certain order, say the resultant equation; and then to each value of x (or y) satisfying this equation there corresponds
in general a single value of y '(or x), and consequently a single point of intersection; the number of intersections is thus equal to the order of the resultant
equation in x (or y).

Supposing that the two curves are of the orders m, n, respectively, then the order of the resultant equation is in general and at most = #m; in particular,
if the curve of the order n is an arbitrary line (=i), then the order of the resultant equation is = w; and the curve of the order m meets therefore the line in
m points. But the resultant equation may have all or any of its roots imaginary, and it is thus not always that there are m real intersections.

The notion of imaginary intersections, thus presenting itself, through algebra, in geometry, must be accepted in geometry and it in fact plays an
all-important part in modern geometry. As in algebra we say that an equation of the wtth order has m roots, viz. we state this generally without in the first
instance, or it may be without ever, distinguishing whether these are real or imaginary; so in geometry we say that a curve of the tntb order is met by an
arbitrary line in m points, or rather we thus, through algebra, obtain the proper geometrical definition of a curve of the mth order, as a curve which is met by
an arbitrary line in m points (that is, of course, in m, and not more than m, points).

The theorem of the m intersections has been stated in regard to an arbitrary line; in fact, for particular lines the resultant equation may be or appear to
be of an order less than m; for instance, taking m=2, if the hyperbola xy 1=0 be cut by the line y=j8, the resultant equation in x is fix 1=0, and there is
apparently only the intersection (x = i//3, y =/3) ; but the theorem is, in fact, true for every line whatever: a curve of the order m meets every line whatever
in precisely m points. We have, in the case just referred to, to take account of a point at infinity on the line y=j3; the two intersections are the point
(x=i/J3, y=fl), and the point at infinity on the line y= /3. .

It is, moreover, to be noticed that the points at infinity may be all or any of them imaginary, and that the points of intersection, whether finite or at
infinity, real or imaginary, may coincide two or more of them together, and have to be counted accordingly; to support the theorem in its universality, it is
necessary to take account of these various circumstances.

5. Line at Infinity. The foregoing notion of a point at infinity is a very important one in modern geometry; and we have also to consider the paradoxical
statement that in plane geometry, or say as regards the plane, infinity is a right line. This admits of an easy illustration in solid geometry. If with a given
centre of projection, by drawing from it lines to every point of a given line, we project the given line on a given plane, the projection is a line, i.e. this
projection is the intersection of the given plane with the plane through the centre and the given line. Say the projection is always a line, then if the figure
is such that the two planes are parallel, the projection is the intersection of the given plane by a parallel plane, or it is the system of points at infinity
on the given plane, that is, these points at infinity are regarded as situate on a given line, the line infinity of the given plane. 1

1 In solid geometry infinity is a plane^ its intersection with any given plane being the right line which is the infinity of this given plane.

Reverting to the purely plane theory, infinity is a line, related like any other right line to the curve, and thus intersecting it in w points, real or
imaginary, distinct or coincident.

Descartes in the Geometric defined and considered the remarkable curves called after him the ovals of Descartes, or simply Cartesians, which will be again
referred to. The next important work, founded on the Geometrie, was Sir Isaac Newton's Enumeratio linearum tertii ordinis (1706), establishing a classification
of cubic curves founded chiefly on the nature of their infinite branches, which was in some details completed by James Stirling (1692-1770), Patrick Murdoch (d.
1774) and Gabriel Cramer; the work also contains the remarkable theorem (to be again referred to), that there are five kinds of cubic curves giving by their
projections every cubic curve whatever. Various properties of curves in general, and of cubic curves, are established in Colin Maclaurin's memoir, "De linearum
geometricarum proprietatibus generalibus Tractatus " (posthumous, say 1746, published in the 6th edition of his Algebra). We have in it a particular kind of
correspondence of two points on a cubic curve, viz. two points correspond to each other when the tangents at the two points again meet the cubic in the same
point.

6. Reciprocal Polars. Intersections of Circles. Duality. Trilinear and Tangential Co-ordinates. The Geometrie descriptive, by Gaspard Monge, was written in
the year 1794 or 1795 (7th edition, Paris, 1847), and in it we have stated, in piano with regard to the circle, and in three dimensions with regard to a surface
of the second order, the fundamental theorem of reciprocal polars, viz. " Given a surface of the second order and a circumscribed conic surface which touches it
... then if the conic surface moves so that its summit is always in the same plane, the plane of the curve of contact passes always through the same point." The
theorem is here referred to partly on account of its bearing on the theory of imaginaries in geometry. It is in Charles Julian Brianchon's memoir " Sur les
surfaces du second degre " (Jour. Polyt. t. vi. 1806) shown how for any given position of the summit the plane of contact is determined, or reciprocally; say
the plane XY is determined when the point P is given, or reciprocally; and it is noticed that when P is situate in the interior of the surface the plane XY does
not cut the surface; that is, we have a real plane XY intersecting the surface in the imaginary curve of contact of the imaginary circumscribed cone having for
its summit a given real point P inside the surface.

Stating the theorem in regard to a conic, we have a real point P (called the pole) and a real line XY (called the polar), the line joining the two (real or
imaginary) points of contact of the (real or imaginary) tangents drawn from the point to the conic; and the theorem is that when the point describes a line the
line passes through a point, this line and point being polar and pole to each other. The term " pole " was first used by Francois Joseph Servois, and " polar "
by Joseph Diez Gergonne (Gerg. t. i. and iii., 1810-1813); and from the theorem we have the method of reciprocal polars for the transformation of geometrical
theorems, used already by Brianchon (in the memoir above referred to) for the demonstration of the theorem called by his name, and in a similar manner by
various writers in the earlier volumes of Gergonne. We are here concerned with the method less in itself than as leading to the general notion of duality.

Bearing in a some'what similar manner also on the theory of imaginaries in geometry (but the notion presents itself in a more explicit form), there is the
memoir by L. Gaultier, on the graphical construction of circles and spheres (Jour. Polyt. t. ix., 1813). The well-known theorem as to radical axes may be stated
as follows. Consider two circles partially drawn so that it does not appear whether the circles, if completed, would or would not intersect in real points, say
two arcs of circles; then we can, by means of a third circle drawn so as to intersect in two real points each of the two arcs, determine a right line, which, if
the complete circles intersect in two real points, passes through the points, and which is on this account regarded as a line passing through two (real or
imaginary) points of intersection of the two circles. The construction in fact is, join the two points in which the third circle meets the first arc, and join
also the two points in which the third circle meets the second arc, and from the point of intersection of the two joining lines, let fall a perpendicular on the
line joining the centre of the two circles; this perpendicular (considered as an indefinite line) is what Gaultier terms the " radical axis of the two circles
"; it is a line determined by a real construction and itself always real; and by what precedes it is the line joining two (real or imaginary, as the case may
be) intersections of the given circles.

The intersections which lie on the radical axis are two out of the four intersections of the two circles. The question as to the remaining two intersections
did not present itself to Gaultier, but it is answered in Jean Victor Poncelet's Traite des propritles projectives (1822), where we find (p. 49) the statement,
"deux circles places arbitrairement sur un plan . . . ont idealement deux points imaginaires communs a 1'infini "; that is, a circle qua curve of the second
order is met by the line infinity in two points; but, more than this, they are the same two points for any circle whatever. The points in question have since
been called (it is believed first by Dr George Salmon) the circular points at infinity, or they may be called the circular points; these are also frequently
spoken of as the points I, J; and we have thus the circle characterized as a conic which passes through the two circular points at infinity; the number of
conditions thus imposed upon the conic is =2, and there remain three arbitrary constants, which is the right number for the circle. Poncelet throughout his work
makes continual use of the foregoing theories of imaginaries and infinity, and also of the before-mentioned theory of reciprocal polars.

Poncelet's two memoirs Sur les centres des moyennes harmoniques and Sur la theorie generale des polaires reciproques, although presented to the Paris Academy
in 1824, were only published (Crelle, t. iii. and iv., 1828, 1829) subsequent to the memoir by Gergonne, Considerations pkilosophiques sur les elemens de la
science de I'etendue (Gerg. t. xvi., 1825-1826). In this memoir by Gergonne, the theory of duality is very clearly and explicitly stated; for instance, we find
" dans la geometric plane, a chaque theoreme il en repond necessairement un autre qui s'en deduit en echangeant simplement entre eux les deux mots points et
droites; tandis que dans la geometric de 1'espace ce sont les mots points et plans qu'il faut echanger entre eux pour passer d'un theoreme a son correlatif ";
and the plan is introduced of printing correlative theorems, opposite to each other, in two columns. There was a reclamation as to priority by Poncelet in the
Bulletin universel reprinted with remarks by Gergonne (Gerg. t. xix., 1827), and followed by a short paper by Gergonne, Rectifications de quelques Ihforemes,
etc., which is important as first introducing the word class. We find in it explicitly the two correlative definitions: " a plane curve is said to be of the
wzth degree (order) when it has with a line m real or ideal intersections," and " a plane curve is said to be of the mth class when from any point of its plane
there can be drawn to it m real or ideal tangents."

It may be remarked that in Poncelet's memoir on reciprocal polars, above referred to, we have the theorem that the number of tangents from a point to a curve
of the order m, or say the class of the curve, is in general and at most m(m-i), and that he mentions that this number is subject to reduction when the curve
has double points or cusps.

The theorem of duality as regards plane figures may be thus stated: two figures may correspond to each other in such manner that to each point and line in
either figure there correspond in the other figure a line and point respectively. It is to be understood that the theorem extends to all points or lines, drawn
or not drawn; thus if in the first figure there are any number of points on a line drawn or not drawn, the corresponding lines in the second figure, produced if
necessary, must meet in a point. And we thus see how the theorem extends to curves, their points and tangents; if there is in the first figure a curve of the
order m, any line meets it in m points; and hence from the corresponding point in the second figure there must be to the corresponding curve m tangents; that
is, the corresponding curve must be of the class m.

Trilinear co-ordinates (see GEOMETRY : Analytical) were firs used by E. E. Bobillier in the memoir Essai sur un nouveau mod de recherche des proprietes de
I'etendue (Gerg. t. xviii., 1827-1828) It is convenient to use these rather than Cartesian co-ordinates We represent a curve of the order m by an equation (*
$x, y, z) m = o the function on the left hand being a homogeneous rational ant integral function of the order m of the three co-ordinates (x, y, z) clearly the
number of constants is the same as for the equation (*$x, y, i) m = o in Cartesian co-ordinates.

The theorem of duality is considered and developed, but chiefly in regard to its metrical applications, by Michel Chasles in the Memoire de geometric sur
deux principes generaux de la science la dualite et I' homo graphic, which forms a sequel to the Aperc.it historique sur I' origine et le developpement des
methodes en geometrii (Mem. de Brux. t. xi., 1837).

We now come to Julius Plucker; his " six equations " were given in a short memoir in Crelle (1842) preceding his great work, the Theorie der algebraischen
Cunien (1844). Plucker first gave a scientific dual definition of a curve, viz.; "A curve is a locus generated by a point, and enveloped by a line the point
moving continuously along the line, while the line rotates continuously about the point " ; the point is a point (ineunt.) of the curve, the line is a tangent
of the curve. And, assuming the above theory of geometrical imaginaries, a curve such that m of its points are situate in an arbitrary line is said to be of the
order m; a curve such that n of its tangents pass through an arbitrary point is said to be of the class n; as already appearing, this notion of the order and
class of a curve is, however, due to Gergonne. Thus the line is a curve of the order i and class o; and corresponding dually thereto, we have the point as a
curve of the order o and class i.

Plucker, moreover, imagined a system of line-co-ordinates (tangential co-ordinates). (See GEOMETRY: Analytical.) The Cartesian co-ordinates (x, y) and
trilinear co-ordinates (x, y, z) are point-co-ordinates for determining the position of a point; the new co-ordinates, say (, TJ, f) are line-co-ordinates for
determining, the position of a line. It is possible, and (not so much for any application thereof as in order to more fully establish the analogy between the
two kinds of co-ordinates) important, to give independent quantitative definitions of the two kinds of co-ordinates; but we may also derive the notion of
line-co-ordinates from that of point-coordinates ; viz. taking r-Hj;y-f-fz=o to be the equation of a line, we say that (, 17, f) are the line-co-ordinates of
this line. A linear relation a+bri+c=o between these co-ordinate determines a point, viz. the point whose point-co-ordinates are (a, b, c); in fact, the
equation in question a+bri+c =o expresses that the equation ^+?jy+fz=o, where (*, y, z) are current point-co-ordinates, is satisfied on writing therein x, y,
z=a, b, c; or that the line in question passes through the point (a, b, c). Thus (, 17, f) are the line-co-ordinates of any line whatever ; but when these,
instead of being absolutely arbitrary, are subject to the restriction a+br)+cf = o, this obliges the line to pass through a point (a, b, c) ; and the
last-mentioned equation a%+bri+c{ =o is considered as the line-equation of this point.

A line has only a point-equation, and a point has only a lineequation; but any other curve has a point-equation and also a line-equation; the point-equation
(*5*, y, z) m =o is the relation which is satisfied by the point-co-ordinates (x, y, z) of each point of the curve; and similarly the line-equation (*J, 77,
f)"=o is the relation which is satisfied by the line-co-ordinates ({, 17, f) of each line (tangent) of the curve.

There is in analytical geometry little occasion for any explicit use of line-co-ordinates; but the theory is very important; it serves to show that in
demonstrating by point-co-ordinates any purely descriptive theorem whatever, we demonstrate the correlative theorem; that is, we do not demonstrate the one
theorem, and then (as by the method of reciprocal polars) deduce from it the other, but we do at one and the same time demonstrate the two theorems; our (x,
y,z.) instead of meaning point-co-ordinates may mean line-co-ordinates, and the demonstration is then in every step of it a demonstration of the correlative
theorem.

Point-singularities

Line-singularities

7. Singularities oj a Curve. Plucker' s Equations. The above dual generation explains the nature of the singularities of a plane curve. The ordinary
singularities, arranged according to a cross division, are

rotation come to rest, and then reverse the direction of its rotation.

2. The node : the point may in the course of its motion come to

coincide with a former position of the point, the two positions of the line not in general coinciding.

4. The double tangent : the line may in the course of its motion

come to coincide with a former position of the line, the two positions of the point not in general coinciding.

It may be remarked that we cannot with a real point and line obtain the node with two imaginary tangents (conjugate or isolated point or acnode), nor again
the real double tangent with two imaginary points of contact; but this is of little consequence, since in the general theory the distinction between real and
imaginary is not attended to.

The singularities (i) and (3) have been termed proper singularities, and (2) and (4) improper; in each of the first-mentioned cases there is a real
singularity, or peculiarity in the motion; in the other two cases there is not; in (2) there is not when the point is first at the node, or when it is secondly
at the node, any peculiarity in the motion; the singularity consists in the point coming twice into the same position; and so in (4) the singularity is in the
line coming twice into the same position. Moreover (i) and (2) are, the former a proper singularity, and the latter an improper singularity, as regards the
motion of the point; and similarly (3) and (4) are, the former a proper singularity, and the latter an improper singularity, as regards the motion of the
line.

But as regards the representation of a curve by an equation, the case is very different.

First, if the equation be in point-co-ordinates, (3) and (4) are in a sense not singularities at all. The curve (*J x, y, z) m = o, or general curve of the
order m, has double tangents and inflections; (2) presents itself as a singularity, for the equations d,(*lx, y, z) m =o,d,(*lx, y, z) m =o, / z (*{ x,y, z)" =
o, implying (*Jj*, y, z) m = o, are not in general satisfied by any values (a, b, c) whatever of (*, y, z), but if such values exist, then the point (a, b, c)
is a node or double point; and (i) presents itself as a Further singularity or sub-case of (2), a cusp being a double point For which the two tangents becomes
coincident.

In line-co-ordinates all is reversed: (i) and (2) are not singularities ; (3) presents itself as a sub-case of (4).

The theory of compound singularities will be referred to farther on.

In regard to the ordinary singularities, we have m, the order, n class,

the whole system being equivalent to three equations only; and it may be added that using a to denote the equal quantities 3/w+i and 3 +*c everything may be
expressed in termsof m,n,a. We have

=o 3,

i = a yn,

25 = m 2 m+8n 30.

2r = n 2 n-\-8m 30.

It is implied in Pliicker's theorem that, m, n, d, c, T, i signifying as above in regard to any curve, then in regard to the reciprocal curve, n, m, T, i, S,
K will have the same significations, viz. for the reciprocal curve these letters denote respectively the order, class, number of nodes, cusps, double tangent
and inflections.

The expression ? m(m+3) S 2n is that of the number of the disposable constants in a curve of the order m with S nodes and K cusps (in fact that there shall
be a node is I condition, a cusp 2 conditions) and the equation (9) thus expresses that the curve and its reciprocal contain each of them the same number of
disposable constants.

For a curve of the order m, the expression $m(m i) S K is termed the " deficiency " (as to this more hereafter) ; the equation (10) expresses therefore that
the curve and its reciprocal have each of them the same deficiency.

With regard to the demonstration of Pliicker's equations it is to be remarked that we are not able to write down the equation in point-co-ordinates of a
curve of the order m, having the given numbers 5 and K of nodes and cusps. We can only use the general equation (*\x, y, z) m = o, say for shortness u=o, of a
curve of the mth order, which equation, so long as the coefficients remain arbitrary, represents a curve without nodes or cusps. Seeking then, for this curve,
the values, n, i, T of the class, number of inflections, and number of double tangents, first, as regards the class, this is equal to the number of tangents
which can be drawn to the curve from an arbitrary point, or what is the same thing, it is equal to the number of the points of contact of these tangents. The
points of contact are found as the intersections of the curve = o by a curve depending on the position of the arbitrary point, and called the "first polar" of
this point; the order of the first polar is =mi, and the number of intersections is thus =m(mi). But it can be shown, analytically or geometrically, that if the
given curve has a node, the first polar passes through this node, which therefore counts as two intersections, and that if the curve has a cusp, the first polar
passes through the cusp, touching the curve there, and hence the cusp counts as three intersections. But, as is evident, the node or cusp is not a point of
contact of a proper tangent from the arbitrary point; we have, therefore, for a node a diminution 2, and for a cusp a diminution 3, in the number of the
intersections; and thus, for a curve with 5 nodes and /c cusps, there is a diminution 25+3/c, and the value of is n = m (m i) 26 3/c.

Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve
called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar ( 8 below) in
regard to the curve breaks up into a pair of lines, and which has an equation H = o, where H is the determinant formed with the second differential coefficients
of in regard to the variables (x, y, z); H=o is thus a curve of the order 3(m 2), and the number of inflections is =^m(m 2). But if the given curve has a node,
then not only the Hessian passes through the node, but it has there a node the two branches at which touch respectively the two branches of the curve; and the
node thus counts as six intersections; so if the curve has a cusp, then the Hessian not only passes through the cusp, but it has there a cusp through which it
again passes, that is, there is a cuspidal branch touching the cuspidal branch of the curve, and besides a simple branch passing through the cusp, and hence the
cusp counts as eight intersections. The node or cusp is not an inflection, and we have thus for a node a diminution 6, and for a cusp a diminution 8, in the
number of the intersections; hence for a curve with 5 nodes and K cusps, the diminution is =66+8*, and the number of inflections is i = $m(m 2) 68 8x.

Thirdly, for the double tangents; the points of contact of these are obtained as the intersections of the curve by a curve II = o, which has not as yet been
geometrically defined, but which is found analytically to be of the order (m 2) (m 2 9); the number of intersections is thus =m(m-2) (mf-g); but if the given
curve has a node then there is a diminution =4(wt 2 m 6), and if it has a cusp then there is a diminution =6(m 2 m 6), where, however, it is to be noticed that
the factor (nfm6) is in the case of a curve having only a node or only a cusp the number of the tangents which can be drawn from the node or cusp to the curve,
and is used as denoting the number of these tangents, and ceases to be the correct expression if the number of nodes and cusps is greater than unity. Hence, in
tke case of a curve which has 5 nodes and K cusps, the apparent diminution 2(m 2 m 6)(25+3*c) is too great, and it has in fact to be diminished by 2 ( (25(8
i)+66/c+f K(K i) j , or the half thereof is 4 for each pair of nodes, 6 for each combination of a node and cusp, and 9 for each pair of cusps. We have thus
finally an expression for 2T, =m (m2) (m 2 9) etc.; or dividing the whole by 2, we have the expression for r given by the third of Pliicker's equations.

It is obvious that we cannot by consideration of the equation u = o in point-co-ordinates obtain the remaining three of Pliicker's equations; they might be
obtained in a precisely analogous manner by means of the equation v = o in line-co-ordinates, but they follow at once from the principle of duality, viz. they
are obtained by the mere interchange of m, B, K, with .n, T, i respectively.

To complete Pliicker's theory it is necessary to take account of compound singularities; it might be possible, but it is at any rate difficult, to effect
this by considering the curve as in course of description by the point moving along the rotating line; and it seems easier to consider the compound singularity
as arising from the variation of an actually described curve with ordinary singularities. The most simple case is when three double points come into
coincidence, thereby giving rise to a triple point; and a somewhat more complicated one is when we have a cusp of the second kind, or node-cusp arising from the
coincidence of a node, a cusp, an inflection, and a double tangent, as shown in the annexed figure, which represents the singularities as on the point of
coalescing. The general conclusion (see Cayley, Quart. Math. Jour. t. vii., 1866, " On the higher singularities of plane curves "; Collected Works, v. 520) is
that every singularity whatever may be considered as compounded of ordinary singularities, say we have a singularity =5' nodes, K 7 cusps, r' double tangents
and i' inflections. So that, in fact, Pliicker's equations properly understood apply to a curve with any singularities whatever. By means of Pliicker's
equations we may form a table:

..(table)..

The table is arranged according to the value of m; and we have m = o, =i, the point; mi, n = o, tfie line; m=2, n=2, the conic; of m = 3, the cubic, there
are three cases, the class being 6, 4 or 3, according as the curve is without singularities, or as it has i node or i cusp; and so of = 4, the quartic, there
are ten cases, where observe that in two of them the class is =6, the reduction of class arising from two cusps or else from three nodes. The ten cases may be
also grouped together into four, according as the number of nodes and cusps (5+x) is =o, i, 2 or 3.

The cases may be divided into sub-cases, by the consideration of compound singularities; thus when m = 4, n = 6, 5 = 3, the three nodes may be all distinct,
which is the general case, or two of them may unite together into the singularity called a tacnode, or all three may unite together into a triple point or else
into an oscnode.

We may further consider the inflections and double tangents, as well in general as in regard to cubic and quartic curves.

The expression for the number of inflections ym(m - 2) for a curve of the order m was obtained analytically by Plucker, but the theory was first given in a
complete form by Hesse in the two papers " t)ber die Elimination, u.s.w.," and " tJber die Wendepuncte der Curven dritter Ordnung " (Crelle, t. xxviii., 1844);
in the latter of these the points of inflection are obtained as the intersections of the curve u o with the Hessian, or curve A = o, where A is the determinant
formed with the second derived functions of u. We have in the Hessian the first instance of a covariant of a ternary form. The whole theory of the inflections
of a cubic curve is discussed in a very interesting manner by means of the canonical form of the equati6n x 3 +y 3 +z +6lxyz= o; and in particular a proof is
given of Pliicker's theorem that the nine points of inflection of a cubic curve lie by threes in twelve lines.

It may be noticed that the nine inflections of a cubic curve represented by an equation with real coefficients are three real, six imaginary; the three real
inflections lie in a line, as was known to Newton and Maclaurin. For an acnodal cubic the six imaginery inflections disappear, and there remain three real
inflections lying in a line. For a crunodal cubic the six inflections which disappear are two of them real, the other four imaginary, and there remain two
imaginary inflections and one real inflection. For a cuspidal cubic the six imaginary inflections and two of the real inflections disappear, and there remains
one real inflection.

A quartic curve has 24 inflections; it was conjectured by George Salmon, and has been verified by H. G. Zeuthen that at most eight of these are real.

The expression %m(m-2)(m?-)) for the number of double tangents of a curve of the order m was obtained by Pliicker only as a consequence of his first, second,
fourth and fifth equations. An investigation by means of the curve II = o, which by its intersections with the given curve determines the points of contact of
the double tangents, is indicated by Cayley, " Recherches sur 1'elimination et la theorie des courbes " (Crelle, t. xxxiv., 1847; Collected Works, vol. i. p.
337), and in part carried out by Hesse in the memoir " Uber Curven dritter Ordnung " (Crelle, t. xxxvi., 1848). A better process was indicated by Salmon in the
" Note on the Double Tangents to Plane Curves," Phil. Mag., 1858; considering the m-2 points in which any tangent to the curve again meets the curve, he showed
how to form the equation of a curve of the order (m-2), giving by its intersection with the tangent the points in question; naking the tangent touch this curve
of the order (m-2), it will be a double tangent of the original curve. See Cayley, " On the Double Tangents of a Plane Curve " (Phil. Trans, t. cxlviii., 1859;
Collected Works, iv. 186), and O. Dersch (Math. Ann. t. vii., 1874). The solution is still in so far incomplete that we have no properties of the curve II = o,
to distinguish one such curve from the several other curves which pass through the points of contact of the double tangents.

A quartic curve has 28 double tangents, their points of contact determined as the intersections of the curve by a curve II = o of the order 14, the equation
of which in a very elegant form was first obtained by Hesse (1849). Investigations in regard to them are given by Plucker in the Theorie der algebraischen
Curven, and in two memoirs by Hesse and Jacob Steiner (Crelle, t. xlv., 1855), in respect to the triads of double tangents which have their points of contact on
a conic and other like relations. It was assumed by Plucker that the number of real double tangents might be 28, 16, 8, 4 or o, but Zeuthen has found that the
last case does not exist.

8. Invariants and Covariants. Polar Curves. The Hessian A has just been spoken of as a covariant of the form u; the notion of invariants and covariants
belongs rather to the form u than to the curve u=o represented by means of this form; and the theory may be very briefly referred to. A curve = o may have some
invariantive property, viz. a property independent of the particular axes of co-ordinates used in the representation of the curve by its equation; for instance,
the curve may have a node, and in order to this, a relation, say A = o, must exist between the coefficients of the equation; supposing the axes of co-ordinates
altered, so that the equation becomes '=o, and writing A' = o for the relation between the new coefficients, then the relations A = o, A' = o, as two different
expressions of the same geometrical property, must each of them imply tiie other; this can only be the case when A, A' are functions differing only by a
constant factor, or say, when A is an invariant of u. If, however, the geometrical property L requires two or more relations between the coefficients, say A =
o, B = o, etc., then we must have between the new coefficients the like relations, A' = o, B' = o, etc., and the two systems of equations must each of them
imply the other; when this is so, the system of equations, A = o, B = o, etc., is said to be invariantive, but it does not follow that A, B, etc., are of
necessity invariants of u. Similarly, if we have a curve U = o derived from the curve u = o in a manner independent of the particular axes of co-ordinates, then
from the transformed equation u' = o deriving in like manner the curve U' = o, the two equations U = o, U' = o must each of them imply the other; and when this
is so, U will be a covariant of u. The case is less frequent, but it may arise, that there are covariant systems U = o, V = o, etc., and U' = o, V' = o, etc.,
each implying the other, but where the functions U, V, etc., are not of necessity covariants of u.

If we take a fixed point (x',y'j?) and a curve = o of order m, and suppose the axes of reference altered, so that x', y', z 7 are linearly transformed in the
same way as the current x, y, z, the curves *V +/~ + z ' T u = o, (r=i, 2, . . .m i)have the covariant property. They are the polar curves of the point with
regard to =o.

The theory of the invariants and covariants of a ternary cubic function u has been studied in detail, and brought into connexion with the cubic curve w=o;
but the theory of the invariants and covariants for the next succeeding case, the ternary quartic function, is still very incomplete.

9. Envelope of a Curve. In further illustration of the Pliickerian dual generation of a curve, we may consider the question of the envelope of a variable
curve. The notion is very probably older, but it is at any rate to be found in Lagrange's Theorie desfonctions cmalyliques (1798) ; it is there remarked that
the equation obtained by the elimination of the parameter a from an equation/ (x,y,a) = o and the derived equation in respect to a is a curve, the envelope of
the series of curves represented by the equation / (x,y,a) = o in question. To develop the theory, consider the curve corresponding to any particular value of
the parameter; this has with the consecutive curve (or curve belonging to the consecutive value of the parameter) a certain number of intersections and of
common tangents, which may be considered as the tangents at the intersections; and the so-called envelope is the curve which is at the same time generated by
the points of intersection and enveloped by the common tangents; we have thus a dual generation. But the question needs to be further examined. Suppose that in
general the variable curve is of the order m with 5 nodes and K cusps, and therefore of the class. with T double tangents and i inflections, m, n, 5, K, r, i
being connected by the Pliickerian equations, the number of nodes or cusps may be greater for particular values of the parameter, but this is a speciality which
may be here disregarded. Considering the variable curve corresponding to a given value of the parameter, or say simply the variable curve, the consecutive curve
has then also 5 and K nodes and cusps, consecutive to those of the variable curve; and it is easy to see that among the intersections of the two curves we have
the nodes each counting twice, and the cusps each counting three times; the number of the remaining intersectiousis rrf 28 3*. Similarly among the common
tangents of the two curves we have the double tangents each counting twice, and the stationary tangents each counting three times, and * the number of the
remaining common tangents is = 2 27 31 ( = m 2 26 3/c, inasmuch as each of these numbers is as was seen =m-\-n). At any one of the m 1 28 3* points the variable
curve and the consecutive curve have tangents distinct from yet infinitesimally near to each other, and each of these two tangents is also infinitesimally near
to one of the w 2 27 31 common tangents of the two curves; whence, attending only to the variable curve, and considering the consecutive curve as coming into
actual coincidence with it, the 2 2731 common tangents are the tangents to the variable curve at the w 2 25 3* points respectively, and the envelope is at the
same time generated by the t 2 25 3* points, and enveloped by the 2 27 31 tangents; we have thus a dual generation of the envelope, which only differs from
Plucker's dual generation, in that in place of a single point and tangent we have the group of w 2 25 3* points and 2 27 31 tangents.

The parameter which determines the variable curve may be given as a point upon a given curve, or say as a parametric point; that is, to the different
positions of the parametric point on the given curve correspond the different variable curves, and the nature of the envelope will thus depend on that of the
given curve; we have thus the envelope as a derivative curve of the given curve. Many well-known derivative curves present themselves in this manner; thus the
variable curve may be the normal (or line at right angles to the tangent) at any point of the given curve; the intersection of the consecutive normals is the
centre of curvature; and we have the evolute as at once the locus of the centre of curvature and the envelope of the normal. It may be added that the given
curve is one of a series of curves, each cutting the several normals at right angles. Any one of these is a " parallel " of the given curve; and it can be
obtained as the envelope of a circle of constant radius having its centre on the given curve. We have in like manner, as derivatives of a given curve, the
caustic, catacaustic or diacaustic as the case may be, and the secondary caustic, or curve cutting at right angles the reflected or refracted rays.

10. Forms of Real Curves. We have in much that precedes disregarded, or at least been indifferent to, reality; it is only thus that the conception of a curve
of the m-th order, as one which is met by every right line in m points, is arrived at; and the curve itself, and the line which cuts it, although both are
tacitly assumed to be real, may perfectly well be imaginary. For real figures we have the general theorem that imaginary intersections, etc., present themselves
in conjugate pairs; hence, in particular, that a curve of an even order is met by a line in an even number (which may be =o) of points; a curve of an odd order
in an odd number of points, hence in one point at least; it will be seen further on that the theorem may be generalized in a remarkable manner. Again, when
there is in question only one pair of points or lines, these, if coincident, must be real; thus, a line meets a cubic curve in three points, one of them real,
and other two real or imaginary; but if two of the intersections coincide they must be real, and we have a line cutting a cubic in one real point and touching
it in another real point. It may be remarked that this is a limit separating the two cases where the intersections are all real, and where they are one real,
two imaginary.

Considering always real curves, we obtain the notion of a branch; any portion capable of description by the continuous motion of a point is a branch; and a
curve consists of one or more branches. Thus the curve of the first order or right line consists of one branch; but in curves of the second order, or conies,
the ellipse and the parabola consist each of one branch, the hyperbola of two branches. A branch is either re-entrant, or it extends both ways to infinity, and
in this case, we may regard it as consisting of two legs (crura, Newton), each extending one way to infinity, but without any definite separation. The branch ,
whether re-entrant or infinite, may have a cusp or cusps, or it may cut itself or another branch, thus having or giving rise to crunodes or double points with
distinct real tangents; an acnode, or double point with imaginary tangents, is a branch by itself, it may be considered as an indefinitely small re-entrant
branch. A branch may have inflections and double tangents, or there may be double tangents which touch two distinct branches; there are also double tangents
with imaginary points of contact, which are thus lines having no visible connexion with the curve. A re-entrant branch not cutting itself may be everywhere
convex, and it is then properly said to be an oval; but the term oval may be used more generally for any re-entrant branch not cutting itself; and we may thus
speak of a once indented, twice indented oval, etc., or even of a cuspidate oval. Other descriptive names for ovals and re-entrant branches cutting themselves
may be used when required; thus, in the last-mentioned case a simple form is that of a figure of eight; such a form may break up into two ovals or into a doubly
indented oval or hour-glass. A form which presents itself is when two ovals, one inside the other, unite, so as to give rise to a crunode in default of a better
name this may be called, after the curve of that name, a limacon . Names may also be used for the different forms of infinite branches, but we have first to
consider the distinction of hyperbolic and parabolic. The leg of an infinite branch may have at the extremity a tangent; this is an asymptote of the curve, and
the leg is then hyperbolic; or the leg may tend to a fixed direction, but so that the tangent goes further and further off to infinity, and the leg is then
parabolic; a branch may thus be hyperbolic or parabolic as to its two legs; or it may be hyperbolic as to one leg and parabolic as to the other. The epithets
hyperbolic and parabolic are of course derived from the conic hyperbola and parabola respectively. The nature of the two kinds of branches is best understood by
considering them as projections, in the same way as we in effect consider the hyperbola and the parabola as projections of the ellipse. If a line fl cut a*n arc
aa' at b, so that the two segments ab, ba' lie on opposite sides of the line, then projecting the figure so that the line Q goes off to infinity, the tangent at
b is projected into the asymptote, and the arc ab is projected into a hyperbolic leg touching the asymptote at one extremity; the arc ba' will at the same time
be projected into a hyperbolic leg touching the same asymptote at the other extremity (and on the opposite side), but so that the two hyperbolic legs may or may
not belong to one and the same branch. And we thus see that the two hyperbolic legs belong to a simple intersection of the curve by the line infinity. Next, if
the line fl touch at b the arc aa' so that the two portions ab, ba' lie on the same side of the line 12, then projecting the figure as before, the tangent at b,
that is, the line 12 itself, is projected to infinity; the arc ab is projected into a parabolic leg, and at the same time the arc ba' is projected into a
parabolic leg, having at infinity the same direction as the other leg, but so that the two legs may or may not belong to the same branch. And we thus see that
the two parabolic legs represent a contact of the line infinity with the curve, the point of contact being of course the point at infinity determined by the
common direction of the two legs. It will readily be understood how the like considerations apply to other cases, for instance, if the line 12 is a tangent at
an inflection, passes through a crunode, or touches one of the branches of a crunode, etc.; thus, if the line 12 passes through a crunode we have pairs of
hyperbolic legs belonging to two parallel asymptotes. The foregoing considerations also show (what is very important) how different branches are connected
together at infinity, and .lead to the notion of a complete branch or circuit.

The two legs of a hyperbolic branch may belong to different asymptotes, and in this case we have the forms which Newton calls inscribed, circumscribed,
ambigene, etc.; or they may belong to the same asymptote, and in this case we have the serpentine form, where the branch cuts the asymptote, so as to touch it
at its two extremities on opposite sides, or the conchoidal form, where it touches the asymptote on the same side. The two legs of a parabolic branch may
converge to ultimate parallelism, as in the conic parabola, or diverge to ultimate parallelism, as in the semi-cubical parabola -f=y?, and the branch is said to
be convergent, or divergent, accordingly; or they may tend to parallelism in opposite senses, as in the cubical parabola y = x i . As mentioned with regard to a
branch generally, an infinite branch of any kind may have cusps, or, by cutting itself or another branch, may have or give rise to a crunode, etc.

ii. Classification of Cubic Curves. We may now consider the various forms of cubic curves as appearing by Newton's Enumeratio, and by the figures belonging
thereto. The species are reckoned as 72, which are numbered accordingly i to 72; but to these should be added 10, 13", 22" and 22 6 . It is not intended here to
consider the division into species, nor even completely that into genera, but only to explain the principle of classification. It may be remarked generally that
there are at most three infinite branches, and that there may besides be a re-entrant branch or oval. The genera may be arranged as follows: 1,2,3,4 redundant
hyperbolas

and thus arranged they correspond to the different relations of the line infinity to the curve. First, if the three intersections by the line infinity are
all distinct, we have the hyperbolas; if the points are real, the redundant hyperbolas, with three hyperbolic branches; but if only one of them is real, the
defective hyperbolas, with one hyperbolic branch. Secondly, if two of the intersections coincide, say if the line infinity meets the curve in a onefold point
and a twofold point, both of them real, then there is always one asymptote: the line infinity may at the twofold point touch the curve, and we have the
parabolic hyperbolas; or the twofold point may be a singular point, viz., a crunode giving the hyperbolisms of the hyperbola; an acnode, giving the hyperbolisms
of the ellipse; or a cusp, giving the hyperbolisms of the parabola. As regards the so-called hyperbolisms, observe that (besides the single asymptote) we have
in, the case of those of the hyperbola two parallel asymptotes; in the case of those of the ellipse the two parallel asymptotes become imaginary, that is, they
disappear; and in the case of those of the parabola they become coincident, that is, there is here an ordinary asymptote, and a special asymptote answering to a
cusp at infinity. Thirdly, the three intersections by the line infinity may be coincident and real; or say we have a threefold point: this may be an inflection,
a crunode or a cusp, that is, the line infinity may be a tangent at an inflection, and we have the divergent parabolas; a tangent at a crunode to one branch,
and we have the trident curve; or lastly, a tangent at a cusp, and we have the cubical parabola.

It is to be remarked that the classification mixes together non-singular and singular curves, in fact, the five kinds presently referred to: thus the
hyperbolas and the divergent parabolas include curves of every kind, the separation being made in the species; the hyperbolisms of the hyperbola and ellipse,
and the trident curve, are nodal; the hyperbolisms of the. parabola, and the cubical parabola, are cuspidal. The divergent parabolas are of five species which
respectively belong to and determine the five kinds of cubic curves; Newton gives (in two short paragraphs without any development) the remarkable theorem that
the five divergent parabolas by their shadows generate and exhibit all the cubic curves.

The five divergent parabolas are curves each of them symmetrical with regard to an axis. There are two non-singular kinds, the one with, the other without,
an oval, but each of them has an infinite (as Newton describes it) campaniform branch ; this cuts the axis at right angles, being at first concave, but
ultimately convex, towards the axis, the two legs continually tending to become at right angles to the axis. The oval may unite itself with the infinite branch,
or it may dwindle into a point, and we have the crunodal and the acnodal forms respectively; or if simultaneously the oval dwindles into a point and unites
itself to the infinite branch, we have the cuspidal form. (See PARABOLA.) Drawing a line to cut any one of these curves and projecting the line to infinity, it
would not be difficult to show how the line should be drawn in order to obtain a curve of any given species. We have herein a better principle of
classification; considering cubic curves, in the first instance, according to singularities, the curves are non-singular, nodal (viz. crunodal or acnodal), or
cuspidal; and we see further that there are two kinds of non-singular curves, the complex and the simplex. There is thus a complete division into the five
kinds, the complex, simplex, crunodal, acnodal and cuspidal. Each singular kind presents itself as a limit separating two kinds of inferior singularity; the
cuspidal separates the crunodal and the acnodal, and these last separate from each other the complex and the simplex.

The whole question is discussed very fully and ably by A. F. Mobius in the memoir " Ueber die Grundformen der Linien dritter Ordnung " (Abh. der K. Sachs.
Ges. zu Leipzig, t. i., 1852). The author considers not only plane curves, but also cones, or, what is almost the same thing, the spherical curves which are
their sections by a concentric Sphere. Stated in regard to the cone, we have there the fundamental theorem that there are two different kinds of sheets; viz.,
the single sheet, not separated into two parts by the vertex (an instance is afforded by the plane considered as a cone of the first order generated by the
motion of a line about a point), and the double or twin-pair sheet, separated into two parts by the vertex (as in the cone of the second order). And it then
appears that there are two kinds of non-singular cubic cones, viz. the simplex, consisting of a single sheet, and the complex, consisting of a single sheet and
a twin-pair sheet; and we thence obtain (as for cubic curves) the crunodal, the acnodal and the cuspidal kinds of cubic cones. It may be mentioned that the
single sheet is a sort of wavy form, having upon it three lines of inflection, and which is met by any plane through the vertex in one or in three lines; the
twin-pair \ sheet has no lines of inflection, and resembles in its form a cone on an oval base.

In general a cone consists of one or more single or twin-pair sheets, and if we consider the section of the cone by a plane, the curve consists of one or
more complete branches, or say circuits, each of them the section of one sheet of the cone; thus, a cone of the second order is one twin-pair sheet, and any
section of it is one circuit composed, it may be, of two branches. But although we thus arrive by projection at the notion of a circuit, it is not necessary to
go out of the plane, and we may (with Zeuthen, using the shorter term circuit for his complete branch) define a circuit as any portion (of a curve) capable of
description by the continuous motion of a point, it being understood that a passage through infinity is permitted. And we then say that a curve consists of one
or more circuits; thus the right line, or curve of the first order, consists of one circuit; a curve of the second order consists of one circuit; a cubic curve
consists of one circuit or else of two circuits.

A circuit is met by any right line always in an even number, or always in an odd number, of points, and it is said to be an even circuit or an odd circuit
accordingly; the right line is an odd circuit, the conic an even circuit. And we have then the theorem, two odd circuits intersect in an odd number of points;
an odd and an even circuit, or two even circuits, in an even number of points. An even circuit not cutting itself divides the plane into two parts, the one
called the internal p'art, incapable of containing any odd circuit, the other called the external part, capable of containing an odd circuit.

We may now state in a more convenient form the fundamental distinction of the kinds of cubic curve. A non-singular cubic is simplex, consisting of one odd
circuit, or it is complex, consisting of one odd circuit and one even circuit. It may be added that there are on the odd circuit three inflections, but on the
even circuit no inflection; it hence also appears that from any point of the odd circuit there can be drawn to the odd circuit two tangents, and to the even
circuit (if any) two tangents, but that from a point of the even circuit there cannot be drawn (either to the odd or the even circuit) any real tangent;
consequently, in a simplex curve the number of tangents from any point is two; but in a complex curve the number is four, or none, four if the point is on the
odd circuit, none if it is on the even circuit. It at once appears from inspection of the figure of a non-singular cubic curve, which is the odd and which the
even circuit. The singular kinds arise as before; in the crunodal and the cuspidal kinds the whole curve is an odd circuit, but in an acnodal kind the acnode
must be regarded as an even circuit.

12. Quartic Curves. The analogous question of the classification of quartics (in particular non-singular quartics and nodal quartics) is considered in
Zeuthen's memoir " Sur les differentes formes des courbes planes du quatrieme ordre " (Math. Ann. t. vii., 1874). A non-singular quartic has only 'even
circuits; it has at most four circuits external to each other, or two circuits one internal to the other, and in this last case the internal circuit has no
double tangents or inflections. A very remarkable theorem is established as to the double tangents of such a quartic : distinguishing as a double tangent of the
first kind a real double tangent which either twice touches the same circuit, or else touches the curve in two imaginary points, the number of the double
tangents of the first kind of a non-singular quartic is = 4; it follows that the quartic has at most 8 real inflections. The forms of the non-singular quartics
are very numerous, but it is not necessary to go further into the question.

We may consider in relation to a curve, not only the line infinity, but also the circular points at infinity; assuming the curve to be real, these present
themselves always conjointly; thus a circle is a conic passing through the two circular points, and is thereby distinguished from other conies. Similarly a
cubic through the two circular points is termed a circular cubic; a quartic through the two points is termed a circular quartic, and if it passes twice through
each of them, that is, has each of them for a node, it is termed a bicircular quartic. Such a quartic is of course binodal (m = 4, 5=2, K=O); it has not in
general, but it may have, a third node or a cusp. Or again, we may have a quartic curve having a cusp at each of the circular points: such a curve is a "
Cartesian," it being a complete definition of the Cartesian to say that it is a bicuspidal quartic curve (m = 4, 5 = o, K=2), having a cusp at each of the
circular points. The circular cubic and the bicircular quartic, together with the Cartesian (being in one point of view a particular case thereof), are
interesting curves which have been much studied, generally, and in reference to their focal properties.

13. Foci. The points called foci presented themselves in the theory of the conic, and were well known to the Greek geometers, but the general notion of a
focus was first established by Plucker (in the memoir " (Jber solche Puncte die bei Curven einer hoheren Ordnung den Brennpuncten der Kegelschnitte entsprechen
" (Crelle, t. x., 1833). We may from each of the circular points draw tangents to a given curve; the intersection of two such tangents (belonging of course to
the two circular points respectively) is a focus. There will be from each circular point X tangents (X, a number depending on the class of the curve and its
relation to the line infinity and the circular points, = 2 for the general conic, i for the parabola, 2 for a circular cubic, or bicircular quartic, etc.); the
X tangents from the one circular point and those from the other circular point intersect in X real foci (viz. each of these is the only real point on each of
the tangents through it), and in X 2 -X imaginary foci; each pair of real foci determines a pair of imaginary foci (the so-called antipoints of the two real
foci), and the jX(X-i) pairs of real foci thus determine the X 2 -X imaginary foci. There are in some cases points termed centres, or singular or multiple foci
(the nomenclature is unsettled), which are the intersections of improper tangents from the two circular points respectively; thus, in the circular cubic, the
tangents to the curve at the two circular points respectively (or two imaginary asymptotes of the curve) meet in a centre.

14. Distance and Angle. Curves described mechanically. The notions of distance and of lines at right angles are connected with the circular points; and
almost every construction of a curve by means of lines of a determinate length, or at right angles to each other, and (as such) mechanical constructions by
means of linkwork, give rise to curves passing the same definite number of times through the two circular points respectively, or say to circular curves, and in
which the fixed centres of the construction present themselves as ordinary, or as singular, foci. Thus the general curve of three bar-motion (or locus of the
vertex of a triangle, the other two vertices whereof move on fixed circles) is a tricircular sextic, having besides three nodes (m = 6, 5 = 3+3+3, = 9), and
having the centres of the fixed circles each for a singular focus; there is a third singular focus, and we have thus the remarkable theorem (due to S. Roberts)
of the triple generation of the curve by means of the three several pairs of singular foci.

Again, the normal, qua line at right angles to the tangent, is connected with the circular points, and these accordingly present themselves in the
before-mentioned theories of evolutes and parallel curves.

15. Theories of Correspondence. We have several recent theories which depend on the notion of correspondence: two points whether in the same plane or in
different planes, or on the same curve or in different curves, may determine each other in such wise that to any given position of the first point there
correspond a' positions of the second point, and to any given position of the second point a positions of the first point; the two points have then an (a, a)
correspondence; and if o, a are each = i, then the two points have a (i, i) or rational correspondence. Connecting with each theory the author's name, the
theories in question are G. F. B. Riemann, the rational transformation of a plane curve; Luigi Cremona, the rational transformation of a plane; and Chasles,
correspondence of points on the same curve, and united points. The theory first referred to, with the resulting notion of " Geschlecht," or deficiency, is more
than the other two an essential part of the theory of curves, but they will all be considered.

Taking u, X, Y, Z to be rational and integral functions (X, Y, Z all of the same order) of the co-ordinates (x, y, z), and u', X', Y', Z' rational and
integral functions (X', Y', Z', all of the same order) of the co-ordinates (x', y', z'), we transform a given curve = o, by the equations of *': y':z' = X: Y:
Z, thereby obtaining a transformed curve u'=o, and a converse set of equations x : y : z=X': Y': Z'; viz. assuming that this is so, the point (x, y, z) on the
curve * = o and the point (*', y', z') on the curve '=o will be points having a (i, i) correspondence. To show how this is, observe that to a given point (x, y,
z) on the curve u = o there corresponds a single point (x', y', z') determined by the equations x': y': z' = X : Y : Z; from these equations and the equation M
= O eliminating x, y, t, we obtain the equation w'=o of the transformed curve. To a gjven point (x', y', z') not on the curve u' = o there corresponds, not a
single point, but the system of points (x, y, z) given by the equations *' : y :*' = X : Y : Z, viz., regarding x', y', z as constants (and to fix the ideas,
assuming that the curves X = o, Y = o, Z = o, have no common intersections), these are the points of intersection of the curves X : Y : Z, = *' : y' : z', but
no one of these points is situate on the curve w = o. If, however, the point (x', y', z') is situate on the curve ' = o, then one point of the system of points
in question is situate on the curve u = o, that is, to a given point of the curve u' = o there corresponds a single point of the curve = o; and hence also this
point must be given by a system of equations such asx : y : z = X' : Y' : Z'.

It is an old and easily proved theorem that, for a curve of the order m, the number 6+/c of nodes and cusps is at most = j(m i) (m 2); for a given curve the
deficiency of the actual number of nodes and cusps below this maximum number, viz. J(m i) (m-2)-d-K, is the " Geschlecht " or "deficiency," of the curve, say
this is = D. When D = o, the curve is said to be unicursal, when =i, bicursal, and so on.

The general theorem is that two curves corresponding rationally to each other have the same deficiency. [In particular a curve and its reciprocal have this
rational or (i, i) correspondence, and it has been already seen that a curve and its reciprocal have the same deficiency.]

A curve of a given order can in general be rationally transformed into a curve of a lower order; thus a curve of any order for which D = o, that is, a
unicursal curve, can be transformed into a line; a curve of any order having the deficiency i or 2 can be rationally transformed into a curve of the order D+2,
deficiency D; and a curve of any order deficience = or3 can be rationally transformed into a curve of the order D+3, deficiency D.

Taking x', y', z' as co-ordinates of a point of the transformed curve, and in its equation writing x' : y' : z' = I : 8: we have a certain irrational
function o f 8, and the theorem is that the co-ordinates x, y, z of any point of the given curve can be expressed as proportional to rational and integral
functions of 6, , that is, of 6 and a certain irrational 'function of 8.

In particular if D=o, that is, if the given curve be unicursal, the transformed curve is a line, is a mere linear function of 0, and the theorem is that the
co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of 6; it is easy to see that for a
given curve of the order m, these functions of must be of the same order m.

If P = l then the transformed curve is a cubic; it can be shown that in a cubic, the axes of co-ordinates being properly chosen, can be expressed as the
square root of a quartic function of 6; and the theorem is that the co-ordinates x, y, z of a point of the bicursal curve can be expressed as proportional to
rational and integral functions of 9, and of the square root of a quartic function of 8.

And so if D2, then the transformed curve is a nodal quartic; can be expressed as the square root of a sextic function of 8 and the theorem is, that the
co-ordinates x, y, z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of 6, and of the square root of a
sextic function of 8. But D=3, we have no longer the like law, viz. t is not expressible as the square root of an octic function of 9.

Observe that the radical, square root of a quartic function, is connected with the theory of elliptic functions, and the radical, square root of a sextic
function, with that of the first kind of Abelian functions, but that the next kind of Abelian functions does not depend on the radical, square root of an octic
function.

It is a form of the theorem for the case D = i, that the coordinates x, y, z of a point of the bicursal curve, or in particular the co-ordinates of a point
of the cubic, can be expressed as proportional to rational and integral functions of the elliptic functions snu, cnu, dnw; in fact, taking the radical to be V i
s . i jW, and writing 0=sn, the radical becomes = cnu, dnu; and we have expressions of the form in question.

It will be observed that the equations x' : y' : z' = X : Y : Z before mentioned do not of themselves lead to the other system of equations * : y : z=X' : Y'
: Z', and thus that the theory does not in anywise establish a (i, i) correspondence between the points (x, y, z) and (x', y', zf) of two planes or of the same
plane; this is the correspondence of Cremona's theory.

In this theory, given in the memoirs " Sulle trasformazioni geometr-che delle figure piani," Mem. di Bologna, t. ii. (1863) and t. v. (1865), we have a
system of equations x' : y' : z'=X : Y : Z which does lead to a system x : y : z = X' : Y' : Z', where, as before, X, Y, Z denote rational and integral
functions, all of the same order, of the co-ordinates x, y, z, and X', Y', Z' rational and integral functions, all of the same order, of the co-ordinates x',
y,' z', and there is thus a (i, i) correspondence given by these equations between the two points (x, y, z) and (x', V, z'). To explain this, observe that
starting from the equations of x' : y 1 : z' = X : Y : Z, to a given point (x, y, z) there corresponds one point (x', y', z') , but that if n be the order of
the functions X, Y, Z, then to a given point x', /, z' there would, if the curves X = o, Y = o, Z = o had no common intersections, correspond n 2 points (x, y,
z). If, however, the functions are such that the curves X = o, Y=o, Z = o have k common intersections, then among the ri 1 points are included these k points,
which are fixed points independent of the point (x', /, z'); so that, disregarding these fixed points, the number of points (*, y, z) corresponding to the given
point (x', y', z') is = n 2 k; and in particular if k = rp I, then we have one corresponding point; and hence the original system of equations x' : y' : z' = X
: Y : Z must lead to the equivalent system x : y : z = X' : Y' : Z'; and in this system by the like reasoning the functions must be such that the curves X' = o,
Y'=o, Z' = o have '* i common intersections. The most simple example is in the two systems of equations x' : y :z"=yz :zx :xy and x :y :z = y'z' : z'x' : x'y';
where yz=o, zx = o, xy = o are conies (pairs of lines) having three common intersections, and where obviously either system of equations leads to the other
system. In the case where X, Y, Z are of an order exceeding 2 the required number n 2 I of common intersections can only occur by reason of common multiple
points on the three curves; and assuming that the curves X=^o, Y = o, Z = o" have CM +03 +03... + o-i common intersections, where the o; points are ordinary
points, the 02 points are double points, the 03 points are triple points, etc., on each curve, we have the condition

01+402+903-)- . . . (n l) 2 o^i = 2 I ; but to this must be joined the condition

01+302+603... + (- i)on_i = n(n+3) -2

(without which the transformation would be illusory); and the conclusion is that 01, o 2 , . . . o,_i may be any numbers satisfying these two equations. It
may be added that the two equations together give

02+303... +J(-i)(w2)o_i = $(n-l)(-2) which expresses that the curves X = o, Y = o, Z=o are unicursal. The transformation may be applied to any curve tt = o,
which is thus rationally transformed into a curve u' =o, by a rational transformation such as is considered in Riemann's theory : hence the two curves have the
same deficiency.

Coming next to Chasles, the principle of correspondence is established and used by him in a series of memoirs relating to the conies which satisfy given
conditions, and to other geometrical questions, contained in the Comptes rendus, t. Iviii. (1864) et seq. The theorem of united points in regard to points in a
right line was given in a paper, June-July 1864, and it was extended to unicursal curves in a paper of the same series (March 1866), " Sur les courbes planes ou
a double courbure dont les points peuvent se determiner individuellement application du principe de correspondance dans la theorie de ces courbes."

The theorem is as follows: if in a unicursal curve two points have an (o, /3) correspondence, then the number of united points (or points each corresponding
to itself) is = o+ 0. In fact in a unicursal curve the co-ordinates of a point are given as proportional to rational and integral functions of a parameter, so
that any point of the curve is determined uniquely by means of this parameter; that is, to each point of the curve corresponds one value of the parameter, and
to each value of the parameter one point on the curve; and the (a, /3) correspondence between the two points is given by an equation of the form (*J0, i)( ,
i)0=o between their parameters 8 and f; at a united point t = 8, and the value of 8 is given by an equation of the order o+/3. The extension to curves of any
given deficiency D was made in the memoir of Cayley, " On the correspondence of two points on a curve," Pore. Land. Math. Soc. t. i. (1866; Collected Works,
vol. vi. p. 9), viz. taking P, P' as the corresponding points in an (a, a') correspondence on a curve of deficiency D, and supposing that when P is given the
corresponding points P' are found as the intersections of the curve by a curve 9 containing the co-ordinates of P as parameters, and having with the given curve
k intersections at the point P, then the number of united points is a = o+o'+2fcD; and more generally, if the curve e intarsect the given curve in a set of
points P' each p times, a set of points Q' each o times, etc., in such manner that the points (P,P') the points (P, Q') etc., are pairs of points corresponding
to each other according to distinct laws; then if (P, P') are points having an (o, a') correspondence with a number = a of united points, (P ,Q') points having
a (/S, /8') correspondence with a number =b of united points, and so on, the theorem is that we have

p(a-a-a')+q(b-/3-0') + . . . = 2kD.

The principle of correspondence, or say rather the theorem of united points, is a most powerful instrument of investigation, which may be used in place of
analysis for the determination of :he number of solutions of almost every geometrical problem. We can by means of it investigate the class of a curve, number of
inflections, etc. in fact, Pliicker's equations; but it is necessary :o take account of special solutions: thus, in one of the most simple instances, in finding
the class of a curve, the cusps present themselves as special solutions.

Imagine a curve of order m, deficiency D, and let the corresponding joints P, P' be such that the line joining them passes through a given point O; this is
an (m I, m i) correspondence, and the value of k is = i, hence the number of united points is = 2m 2+2D; the united points are the points of contact of the
tangents from O and (as special solutions) the cusps, and we have thus the relation n+K = 2m 2+2D; or, writing D = \(m i)(m 2) -/, this is n = m(m i) 26 $*.,
which is right.

The principle in its original form as applying to a right line was used throughout by Chasles in the investigations on the number of the conies which satisfy
given conditions, and on the number of solutions of very many other geometrical problems.

There is one application of the theory" of the (a, a') correspondence between two planes which it is proper to notice.

Imagine a curve, real or imaginary, represented by an equation (involving, it may be, imaginary coefficients) between the Cartesian co-ordinates u, u ' ;
then, writing u = x+iy, u'=x'+iy', the equation determines real values of (x, y), and of (x', y'), corresponding to any given real values of (x', y') and (x, y)
respectively ; that is, it establishes a real correspondence (not of course a rational one) between the points (x, y) and (x', y') ; for example in the
imaginary circle u 1 +u' 2 = (a+bi) t , the correspondence is given by the two equations x?-y*+x'*y* = a*P, xy+x'y' = ab. We have thus a means of geometrical
representation for the portions, as well imaginary as real, of any real or imaginary curve. Considerations such as these have been used for determining the
series of values of the independent variable, and the irrational functions thereof in the theory of Abelian integrals, but the theory seems to be worthy of
further investigation.

1 6. Systems of Curves satisfying Conditions. The researches of Chasles (Comptes Rendus, t. Iviii., 1864, et seq.) refer to the conies which satisfy given
conditions. There is an earlier paper by J. P. E. Fauque de Jonquieres, " Theoremes generaux concernant les courbes geometriques planes d'un ordre quelconque,"
Liouv. t. vi. (1861), which establishes the notion of a system of curves (of any order) of the index N, viz. considering the curves of the order n which satisfy
\n(n+^) i conditions, then the index N is the number of these curves which pass through a given arbitrary point. But Chasles in the first of his papers
(February 1864), considering the conies which satisfy four conditions, establishes the notion of the two characteristics (p,v)oi such a system of conies, viz. n
is the number of the conies which pass through a given arbitrary point, and v is the number of the conies which touch a given arbitrary lin?. And he gives the
theorem, a system of conies satisfying four conditions, and having the characteristics (ju, v) contains 2v p line-pairs (that is, conies, each of them a pair of
lines), and 2/j, v point-pairs (that is, conies, each of them a pair of points, coniques infmiment aplaties), which is a fundamental one in the theory. The
characteristics of the system can be determined when it is known how many there are of these two kinds of degenerate conies in the system, and how often each is
to be counted. It was thus that Zeuthen (in the paper Nyt Bydrag, " Contribution to the Theory of Systems of Conies which satisfy four Conditions " (Copenhagen,
1865), translated with an addition in the Nouvelles Annalcs) solved the question of finding the characteristics of the systems of conies which satisfy four
conditions of contact with a given curve or curves; and this led to the solution of the further problem of finding the number of the conies which satisfy five
conditions of cpntact with a given curve or curves (Cayley, Comptes Rendus, t. Ixiii., 1866; Collected Works, vol. v. p. 542), and " On the Curves which satisfy
given Conditions " (Phil. Trans, t. clviii., 1868; Collected Works, vol. vi. p. 191).

It may be remarked that although, as a process of investigation, it is very convenient to seek for the characteristics of a system of conies satisfying 4
conditions, yet what is really determined is in every case the number of the conies which satisfy 5 conditions; the characteristics of the system (4/) of the
conies which pass through 4/ points are ($p), (4p, U)i the number of the conies which pass through 5 points, and which pass through 4 points and touch i line:
and so in other cases. Similarly as regards cubics, or curves of any other order: a cubic depends on 9 constants, and the elementary problems are to find the
number of the cubics (9/), (8/, i/), etc., which pass through 9 points, pass through 8 points and touch i line, etc. ; but it is in the investigation convenient
to seek for the characteristics of the systems of cubics (8p), etc., which satisfy 8 instead of 9 conditions.

The elementary problems in regard to cubics are solved very completely by S.Maillard in his These, Recherche des caracttristiques des systemes elementaires
des courbes planes du troisieme ordre [Paris, 1871). Thus, considering the several cases of a cubic

1. With a given cusp

2. cusp on given line

3. cusp , .

4. a given node

5. node on given line

6. node

No. of consts.

7. non-singular

be determines in every case the characteristics (ju, v) of the corresponding systems of cubics (4p), ($p, il), etc. The same problems, or most of them, and
also the elementary problems in regard to quartics are solved by Zeuthen, who in the elaborate memoir " Almindelige Egenskaber, etc.," Danish Academy, t. x.
(1873), considers the problem in reference to curves of any order, and applies his results to cubic and quartic curves.

The methods of Maillard and Zeuthen are substantially identical; in each case the question considered is that of finding the characteristics 0*, v) of a
system of curves by consideration of the special or degenerate forms of the curves included in the system. The quantities which have to be 'considered are very
numerous. Zeuthen in the case of curves of any given order establishes between the characteristics /t, v, and 18 other quantities, in all 20 quantities, a set
of 24 equations (equivalent to 23 independent equations), involving(besides the 20 quantities) other quantities relating to the various forms of the degenerate
curves, which supplementary terms he determines, partially for curves of any order, but completely only for quartic curves. It is the discussion and complete
enumeration of the special or degenerate forms of the curves, and of the supplementary terms to which they give rise, that the great difficulty of the question
seems to consist; it would appear that the 24 equations are a complete system, and that (subject to a proper determination of the supplementary terms) they
contain the solution of the general problem.

17. Degeneration of Curves. The remarks which follow have reference to the analytical theory of the degenerate curves which present themselves in the
foregoing problem of the curves which satisfy given conditions.

A curve represented by an equation in point-co-ordinates may break up: thus if Pi, P 2l ... be rational and integral functions of the co-ordinates (x,y,z) of
the orders m\, mt... respectively, we have the curve Pi 01 Pi 02 . . . =o, of the order m, =aifn 1 + a 2 m 2 + .... composed of the curve PI =o taken ai times,
the curve P 2 = o taken a 2 times, etc.

Instead of the equation Pi al P 2 u ... =o, we may start with an equation = o, where u is a function of the order TO containing a parameter 8, and for a
particular value say 6 = 0, of the parameter reducing itself to Pi 1 P 2 a2 . . . . Supposing 6 indefinitely small, we have what may be called the penultimate
curve, and when 6 the ultimate curve. Regarding the ultimate curve as derived from a given penultimate curve, we connect with the ultimate curve.^and consider
as belonging to it, certain points called " summits " on the component curves Pi = o, P 2 = o respectively; a summit Z is a point such that, drawing from an
arbitrary point O the tangents to the penultimate curve, we have OS as the limit of one of these tangents. The ultimate curve together with its summits may be
regarded as a degenerate form of the curve = o. Observe that the positions of the summits depend on the penultimate curve = o, viz. on the values of the
coefficients in the terms multiplied by 0, O 2 ,...; they are thus in some measure arbitrary points as regards the ultimate curve Pi al P 2 u . . . = o.

It may be added that we have summits only on the component curves PI=O, of a multiplicity oii; the number of_ summits on such a curve is in general = (o t 1
-ai)ii 1 . Thus assuming that the penultimate curve is without nodes or cusps, the number of the tangents to it is = w 2 m, = (oiWii-l-aaWa-l-...)* (fliWJi
-|-fliWXi -r ) Taking PI = O to have 81 nodes and K, cusps, and therefore its class n, to be = Wi 2 m: 28; SKI, etc., the expression for the number of tangents
to the penultimate curve is

(o 2 J -o 2 )m 2 I + . . .

where a term 2oinmim 2 indicates tangents which are in the limit the lines drawn to the intersections of the curves Pi =o, P = o each line 2oiu times; a term
o 1 (ni+28i+3it 1 ) tangents which are in the limit the proper tangents to Pi=o each 01 times, the lines to its nodes each 2n times, and the lines to its cusps
each 301 times the remaining terms (oi 2 ai)mi 2 + (o 2 2 aa)mi'+ . . . indicate tangents which are in the limit the lines drawn to the several summits, that is
we have (ai 2 ai)mi 2 summits on the curve Pi=O, etc.

There is, of course, a precisely similar theory as regards line co-ordinates; taking Ed, n 2 , etc., to be rational and integral functions of the
co-ordinates (f , ;, f ) we connect with the ultimate curve ni^Hj 02 . . . =0, and consider as belonging to it, certain lines, which for the moment may be
called " axes " tangents to the component curves IIi = pi, 1X2 = respectively. Considering an equation in ooint-co-ordinates, we may have among the component
curves right lines, and if in order to put these in evidence we take the equation to be LiT 1 . . Pi al ... =o, where Li =o is a right line, Pi =o a curve oi
the second or any higher order, then the curve will contain as part of itself summits not exhibited in this equation, but the corresponding line-equation will
be ) A sl ...IIi al =o, where Ai=o, ...are the equations of the summits in question, IIi=o, etc., are the lineequations corresponding to the several
point-equations Pi =o, etc. ; and this curve will contain as part of itself axes not exhibited by this equation, but which are the lines Li = o,...of the
equation in point-co-ordinates.

1 8. Twisted Curves. In conclusion a little may be said as to curves of double curvature, otherwise twisted curves or curves in space. The analytical theory
by Cartesian co-ordinates was first considered by Alexis Claude Clairaut, Recherches sur les courbes a double courbure (Paris, 1731). Such a curve may be
considered as described by a point, moving in a line which at the same time rotates about the point in a plane which at the same time rotates about the line;
the point is a point, the line a tangent, and the plane an osculating plane, of the curve; moreover the line is a generating line, and the plane a tangent
plane, of a developable surface or torse, having the curve for its edge of regression. Analogous to the order and class of a plane curve we have the order, rank
and class of the system (assumed to be a geometrical one), viz. if an arbitrary plane contains m points, an arbitrary line meets r lines, and an arbitrary point
lies in n planes, of the system, then m, r, n are the order, rank and class respectively. The system has singularities, and there exist between m, r, n and the
numbers of the several singularities equations analogous to Pliicker's equations for a plane curve.

It is a leading point in the theory that a curve in space cannot in general be represented by means of two equations U=o, V=o; the two equations represent
surfaces, intersecting in a curve; but there are curves which are not the complete intersection of any two surfaces; thus we have the cubic in space, or skew
cubic, which is the residual intersection of two quadric surfaces which have a line in common; the equations U=o, V=o of the two quadric surfaces represent the
cubic curve, not by itself, but together with the line.